10.2 Permutations and applications  (Page 2/2)

Permutation with repetition

When order matters and an object can be chosen more than once then the number of

permutations is:

where $n$ is the number of objects from which you can choose and $r$ is the number to be chosen.

For example, if you have the letters A, B, C, and D and you wish to discover the number of ways of arranging them in three letter patterns (trigrams) you find that there are $4^3$ or 64 ways. This is because for the first slot you can choose any of the four values, for the second slot you can choose any of the four, and for the final slot you can choose any of the four letters. Multiplying them together gives the total.

Applications

The binomial theorem

In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version reads

Whenever $n$ is a positive integer, the numbers

are the binomial coefficients (the coefficients in front of the powers).

For example, here are the cases n = 2, n = 3 and n = 4:

The coefficients form a triangle, where each number is the sum of the two numbers above it:

This formula, and the triangular arrangement of the binomial coefficients, are often attributed to Blaise Pascal who described them in the 17th century. It was, however, known to the Chinese mathematician Yang Hui in the 13th century, the earlier Persian mathematician Omar Khayyám in the 11th century, and the even earlier Indian mathematician Pingala in the 3rd century BC.

Exercises

1. Tshepo and Sally go to a restaurant, where the menu is:

   Starter	Main Course	Dessert
   Chicken wings	Beef burger	Chocolate ice cream
   Mushroom soup	Chicken burger	Strawberry ice cream
   Greek salad	Chicken curry	Apple crumble
   	Lamb curry	Chocolate mousse
   	Vegetable lasagne

   1. How many different combinations (of starter, main course, and dessert) can Tshepo have?
   2. Sally doesn't like chicken. How many different combinations can she have?
2. Four coins are thrown, and the outcomes recorded. How many different ways are there of getting three heads? First write out the possibilities, and then use the formula for combinations.
3. The answers in a multiple choice test can be A, B, C, D, or E. In a test of 12 questions, how many different ways are there of answering the test?
4. A girl has 4 dresses, 2 necklaces, and 3 handbags.
   1. How many different choices of outfit (dress, necklace and handbag) does she have?
   2. She now buys two pairs of shoes. How many choices of outfit (dress, necklace, handbag and shoes) does she now have?
5. In a soccer tournament of 9 teams, every team plays every other team.
   1. How many matches are there in the tournament?
   2. If there are 5 boys' teams and 4 girls' teams, what is the probability that the first match will be played between 2 girls' teams?
6. The letters of the word 'BLUE' are rearranged randomly. How many new words (a word is any combination of letters) can be made?
7. The letters of the word 'CHEMISTRY' are arranged randomly to form a new word. What is the probability that the word will start and end with a vowel?
8. There are 2 History classes, 5 Accounting classes, and 4 Mathematics classes at school. Luke wants to do all three subjects. How many possible combinations of classes are there?
9. A school netball team has 8 members. How many ways are there to choose a captain, vice-captain, and reserve?
10. A class has 15 boys and 10 girls. A debating team of 4 boys and 6 girls must be chosen. How many ways can this be done?
11. A secret pin number is 3 characters long, and can use any digit (0 to 9) or any letter of the alphabet. Repeated characters are allowed. How many possible combinations are there?